### A constrained triplet: *I*, *V* and *R*

Potential difference, current and resistance are constrained. All three are interlinked, in that changing any one is likely to change at least one of the others–but which one, and how?

You need to do some careful physical reasoning about this question to get sensible answers. Take particular care to identify the resistance, voltage and current under consideration.

Let's start with a complete circuit, as this is what we have studied so far. We have already discussed this: just consider what you do in setting up the circuit. Clearly the current responds to changes in the other two. The resistance and the potential difference jointly set the current. There is only one current, one resistance and one potential difference. Changing either the potential difference or the resistance, or both, (very nearly) instantaneously changes the current–certainly the change is faster than you can measure with school meters. The change is not a straightforward jump from one state to another–there is a bit of a fluctuation, as the shock wave caused by the change ripples around the circuit (at nearly the speed of light; the teaching approach using the big circuit in the SPT: Electric circuits topic emphasises this speed of change). So, *I* = *V**R* is a connection that is true all at one time. It is not a case of first one, then the other in this relationship. The three quantities are constrained, so you can only change the current through changes to either of the other two quantities.

### Calculating for a resistor that is a part of a loop

But what about doing calculations for a resistor that is just a part of a circuit loop? First use the relationship *I* = *V**R* for the complete loop to find the current (we'll always use simple

for the resistance of the complete loop and *R*

for the current in that loop). We'll leave the details of how to calculate the value of *I**R* from many resistors until later. For now it's important to get the idea that one resistor can replace many, without changing the current in the loop.

Then focus on a resistor that is one of many in a loop. You have chosen the resistor and the current is known. So now use the all-at-one-time connection linking *R*_{1}, *V*_{1} and *I* arranged as *V*_{1} = *R*_{1} × I to find the potential difference across this one resistor. We'll always use *V*_{1}, *V*_{2}, etc and *R*_{1}, *R*_{2}, etc for potential differences and resistances that contribute to a whole loop. We suggest that you are also consistent about this, to help keep track of what is being discussed.

So you need to take care to deal either with the whole loop, using *I* = *V**R*, or with just a component in the loop, in which *V*_{1} = *R*_{1} × *I* is often useful.

Here's a sample calculation, laid out:

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